When I also give the switch 12v to those same poles, then connect seperate wires to the arduino (via a 10k resistor) I can no longer read the state.
Can you draw a diagram of that condition I am having trouble turning your words to a diagram in my head. Applying more than 5V to an Arduino input will damage it.
You talk of "The switch has three terminals" but then say "just connecting the two poles of the button", is a terminal the same as a pole or what?
You just need to sense the output voltage via a resistor-divider. Don't treat it like a push button, this
kind of switch does not switch to ground.
Most likely one pin is ground. One pin is the supply pin for the light and one pin is the switched pin, shorting to GND when pressed. You probably wired it incorrectly. Give us more info about the switch and we might be able to help better.
In general, for switching an unknown load, a relay gives good isolation .
But more detail would be useful for more specific advice.